In combustion theory, a thin flame zone is usually replaced by a free interface.
A very challenging problem is the derivation of a self-consistent equation for the
flame front which yields a reduction of the dimensionality of the system. A paradigm
is the Kuramoto-Sivashinsky (K-S) equation, which models cellular instabilities and
turbulence phenomena. In this survey paper, we browse through a series of models in
which one reaches a fully nonlinear parabolic equation for the free interface, involving
pseudo-differential operators. The K-S equation appears to be asymptotically the
lowest order of approximation near the threshold of stability.